AT-stable stochastic binary layer

Integrate this into a stochastic binary recurrent or state-space layer, not into the optimizer. Let the hidden state have width N and let x_t be the external input. Parameterize an interaction matrix G with entries scaled as G_ij/sqrt(2N), a positive gain beta, and input bias b_t=W_x x_t+b_0. At each time step compute u_t=b_t+beta G s_{t-1}/sqrt(2N), means m_t=tanh(u_t), and stochastic binary states s_t in {-1,+1}^N using P(s_{t,i}=1)=(1+m_{t,i})/2. During training, compute q_hat_t=N^{-1} sum_i m_{t,i}^2 and chi_hat_t=beta^2 N^{-1} sum_i sech^4(u_{t,i}); add the displayed AT penalty averaged over time. Alternatively, use a hard controller after every optimizer step: beta <- beta exp[-eta_beta max(0, chi_hat-(1-delta))], with stop-gradient through chi_hat. For a replica diagnostic, run two stochastic copies with the same G and input sequence, calculate R_t=N^{-1} sum_i s_{t,i}^{(1)}s_{t,i}^{(2)}, and track its variance. The paper supplies the scalar response criterion and predicts overlap concentration only under SK-like mean-field assumptions, so estimate all neural quantities empirically. First test a width-256 tanh or binary vanilla RNN on sequential MNIST or the adding task, comparing unconstrained gain, spectral normalization, and AT regularization at equal parameter count. Use eight stochastic replicas and measure gradient-norm variance, hidden-state Lyapunov growth, replica-overlap variance, task loss, and performance at sequence lengths two and four times the training length. Success means lower long-horizon loss variance and bounded overlap variance without sacrificing short-horizon accuracy; a useful target is at least a twofold reduction in replica-overlap or gradient variance at equal validation loss.